A269549 Expansion of (-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1).

1, -199, -19799, -1940399, -190139599, -18631740599, -1825720439399, -178901971320799, -17530567468999199, -1717816709990600999, -168328507011609898999, -16494475870427779501199, -1616290306794910781218799, -158379955590030828779941399, -15519619357516226309653038599

Offset: 0

Author

Michel Marcus, Feb 29 2016

Keywords

• sign
• easy

Comments

Mc Laughlin (2010) gives an identity relating ten sequences, denoted a_k, b_k, ..., f_k, p_k, q_k, r_k, s_k. This is the sequence c_k.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• J. Mc Laughlin, An identity motivated by an amazing identity of Ramanujan, Fib. Q., 48 (No. 1, 2010), 34-38.
• Index entries for linear recurrences with constant coefficients, signature (99,-99,1).

Crossrefs

Cf. A010701, A261004, A269548, A269550, A269551, A269552, A269553, A269554, A269555, A269556.

Programs

Magma

m:=20; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((-x^2+298*x-1)/(x^3-99*x^2+99*x-1))); // Bruno Berselli, Mar 01 2016

Mathematica

CoefficientList[Series[(-x^2 + 298 x - 1)/(x^3 - 99 x^2 + 99 x - 1), {x, 0, 20}], x] (* or *) Table[FullSimplify[37/12 + ((2 Sqrt[6] - 5)/(2 Sqrt[6] + 5)^(2 n) - (2 Sqrt[6] + 5) (2 Sqrt[6] + 5)^(2 n)) 5/24], {n, 0, 20}] (* Bruno Berselli, Mar 01 2016 *)

PARI

Vec((-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1) + O(x^20))

Sage

gf = (-x^2+298*x-1)/(x^3-99*x^2+99*x-1)
print(taylor(gf, x, 0, 20).list()) # Bruno Berselli, Mar 01 2016

Formula

G.f.: (-x^2 + 298*x - 1)/(x^3 - 99*x^2 + 99*x - 1).

a(n) = 37/12 + ((2*sqrt(6) - 5)/(2*sqrt(6) + 5)^(2*n) - (2*sqrt(6) + 5)*(2*sqrt(6) + 5)^(2*n))*5/24. - Bruno Berselli, Mar 01 2016